Section2.11Variables, Expressions, and Equations Chapter Review

Subsection2.11.1Variables and Evaluating Expressions

In Section 2.1 we covered the definitions of variables and expressions, and how to evaluate an expression with a particular number. We learned the formulas for perimeter and area of rectangles, triangles, and circles.

Evaluating Expressions

When we evaluate an expression's value, we substitute each variable with its given value.

Example2.11.4

This expression has four terms: \(5x\text{,}\) \(-3x^2\text{,}\) \(2x\text{,}\) and \(5x^2\text{.}\) Both \(5x\) and \(2x\) are like terms; also \(-3x^2\) and \(5x^2\) are like terms. When we combine like terms, we get:

\begin{equation*}
5x-3x^2+2x+5x^2=7x+2x^2
\end{equation*}

Note that we cannot combine \(7x\) and \(2x^2\) because \(x\) and \(x^2\) represent different quantities.

Subsection2.11.4Equations and Inequalities as True/False Statements

In Section 2.4 we covered the definitions of an equation and an inequality, as well as how to verify if a particular number is a solution to them.

Checking Possible Solutions

Given an equation or an inequality (with one variable), checking if some particular number is a solution is just a matter of replacing the value of the variable with the specified number and determining if the resulting equation/inequality is true or false. This may involve some amount of arithmetic simplification.

Subsection2.11.5Solving One-Step Equations

In Section 2.5 we covered to to add, subtract, multiply, or divide on both sides of an equation to isolate the variable, summarized in Fact 2.5.12. We also learned how to state our answer, either as a solution or a solution set. Last we discussed how to solve equations with fractions.

Subsection2.11.6Solving One-Step Inequalities

Solving One-Step Inequalities

When we solve linear inequalities, we also use Properties of Equivalent Equations with one small complication: When we multiply or divide by the same negative number on both sides of an inequality, the direction reverses!

Example2.11.7

Solve the inequality \(-2x\geq12\text{.}\) State the solution set with both interval notation and set-builder notation.

In summary, \(1995.98\) is approximately \(85.07\%\) of \(2346.19\text{.}\)

Subsection2.11.8Modeling with Equations and Inequalities

In Section 2.8 we covered how to translate phrases into mathematics, and how to set up equations and inequalities for application models.

Modeling with Equations and Inequalities

To set up an equation modeling a real world scenario, the first thing we need to do is to identify what variable we will use. The variable we use will be determined by whatever is unknown in our problem statement. Once we've identified and defined our variable, we'll use the numerical information provided in the equation to set up our equation.

Example2.11.9

A bathtub contains 2.5 ft3 of water. More water is being poured in at a rate of 1.75 ft3 per minute. When will the amount of water in the bathtub reach 6.25 ft3?

Since the question being asked in this problem starts with “when,” we immediately know that the unknown is time. As the volume of water in the tub is measured in ft3 per minute, we know that time needs to be measured in minutes. We'll defined \(t\) to be the number of minutes that water is poured into the tub. Since each minute there are 1.75 ft3 of water added, we will add the expression \(1.75t\) to \(2.5\) to obtain the total amount of water. Thus the equation we set up is:

Subsection2.11.11Exercises

1

A trapezoid’s area can be calculated by the formula \(A=\frac{1}{2}(b_1+b_2)h\text{,}\) where \(A\) stands for area, \(b_1\) for the first base’s length, \(b_2\) for the second base’s length, and \(h\) for height.

2

A trapezoid’s area can be calculated by the formula \(A=\frac{1}{2}(b_1+b_2)h\text{,}\) where \(A\) stands for area, \(b_1\) for the first base’s length, \(b_2\) for the second base’s length, and \(h\) for height.

31

The highlighted row means each serving of macaroni and cheese in this box contains \({7\ {\rm g}}\) of fat, which is \(14\%\) of an average person’s daily intake of fat. What’s the recommended daily intake of fat for an average person?

The recommended daily intake of fat for an average person is .

32

The following is a nutrition fact label from a certain macaroni and cheese box.

The highlighted row means each serving of macaroni and cheese in this box contains \({5.5\ {\rm g}}\) of fat, which is \(10\%\) of an average person’s daily intake of fat. What’s the recommended daily intake of fat for an average person?

The recommended daily intake of fat for an average person is .

33

Jerry used to make \(13\) dollars per hour. After he earned his Bachelor’s degree, his pay rate increased to \(48\) dollars per hour. What is the percentage increase in Jerry’s salary?

The percentage increase is .

34

Eileen used to make \(14\) dollars per hour. After she earned her Bachelor’s degree, her pay rate increased to \(49\) dollars per hour. What is the percentage increase in Eileen’s salary?

The percentage increase is .

35

After a \(10\%\) increase, a town has \(550\) people. What was the population before the increase?

Before the increase, the town’s population was .

36

After a \(70\%\) increase, a town has \(1020\) people. What was the population before the increase?

Before the increase, the town’s population was .

37

A bicycle for sale costs \({\$254.88}\text{,}\) which includes \(6.2\%\) sales tax. What was the cost before sales tax?

Assume the bicycle’s price before sales tax is \(p\) dollars. Write an equation to model this scenario. There is no need to solve it.

38

A bicycle for sale costs \({\$283.77}\text{,}\) which includes \(5.1\%\) sales tax. What was the cost before sales tax?

Assume the bicycle’s price before sales tax is \(p\) dollars. Write an equation to model this scenario. There is no need to solve it.

39

The property taxes on a \(2100\)-square-foot house are \({\$4{,}179.00}\) per year. Assuming these taxes are proportional, what are the property taxes on a \(1700\)-square-foot house?

Assume property taxes on a \(1700\)-square-foot house is \(t\) dollars. Write an equation to model this scenario. There is no need to solve it.

40

The property taxes on a \(1600\)-square-foot house are \({\$1{,}600.00}\) per year. Assuming these taxes are proportional, what are the property taxes on a \(2000\)-square-foot house?

Assume property taxes on a \(2000\)-square-foot house is \(t\) dollars. Write an equation to model this scenario. There is no need to solve it.

41

A swimming pool is being filled with water from a garden hose at a rate of \(5\) gallons per minute. If the pool already contains \(30\) gallons of water and can hold \(135\) gallons, after how long will the pool overflow?

Assume \(m\) minutes later, the pool would overflow. Write an equation to model this scenario. There is no need to solve it.

42

A swimming pool is being filled with water from a garden hose at a rate of \(8\) gallons per minute. If the pool already contains \(40\) gallons of water and can hold \(280\) gallons, after how long will the pool overflow?

Assume \(m\) minutes later, the pool would overflow. Write an equation to model this scenario. There is no need to solve it.

43

Use the commutative property of addition to write an equivalent expression to \({5b+31}\text{.}\)

44

Use the commutative property of addition to write an equivalent expression to \({6q+79}\text{.}\)

45

Use the associative property of multiplication to write an equivalent expression to \({3\!\left(4r\right)}\text{.}\)

46

Use the associative property of multiplication to write an equivalent expression to \({4\!\left(7m\right)}\text{.}\)

47

Use the distributive property to write an equivalent expression to \({10\!\left(p+2\right)}\) that has no grouping symbols.

48

Use the distributive property to write an equivalent expression to \({7\!\left(q+6\right)}\) that has no grouping symbols.

49

Use the distributive property to simplify \({4+9\!\left(2+4y\right)}\) completely.

50

Use the distributive property to simplify \({9+4\!\left(9+3r\right)}\) completely.

51

Use the distributive property to simplify \({6-4\!\left(1-6a\right)}\) completely.

52

Use the distributive property to simplify \({3-9\!\left(-5-6b\right)}\) completely.